Basic Concepts of Fractions and its...

Simplifying and Finding Equivalent Fractions

After studying this section, you will be able to: -

1. Understand and use some basic mathematical definitions.

2. Simplify fractions to lowest terms.

3. Change forms between improper fractions and mixed numbers.

4. Change a fraction into an equivalent one with a different denominator.

This tutorial is designed to give you a mental ‘‘warm-up.” In this chapter you’ll be able to step back a bit and tone up your math skills. This brief review of arithmetic will increase your math flexibility and give you a good running start into algebra.

Basic Definitions

Whole numbers are the set of numbers 0, 1, 2, 3, 4, 5, 6, 7.... They are used to describe whole objects, or entire quantities.

Fractions are a set of numbers that are used to describe parts of whole quantities. In the object shown below there are four equal parts. The three of the four parts that are shaded are represented by the fraction 3/4. In the fraction 3/4 the number 3 is called the numerator and the number 4, the denominator.
3
4
3. <— Numerator is on the top
4 <— Denominator is on the bottom

The denominator of a fraction shows the number of equal parts in the whole and the
numerator shows the number of these parts being talked about or being used.

Numerals are symbols we use to name numbers. There are many different numerals that can be used to describe the same number. We know that 1/2 = 2/4. The fractions 1/2 and 2/4 both describe the same number.

Usually, we find it more useful to use fractions that are simplified. A fraction is considered to be in simplest form when the numerator (top) and the denominator (bottom) can both be divided exactly by no number other than 1.

1/2 is in simplest form.

2/4 is not in simplest form since the numerator and the denominator can both be divided by 2.

If you get the answer 2/4 to a problem, you should state it in simplest form, 1/2. The process of changing 2/4 to 1/2 is called simplifying the fraction.

Simplifying Fractions

Natural numbers or counting numbers are the set of whole numbers excluding 0. Thus the natural numbers are the numbers 1, 2, 3, 4, 5, 6....

When two or more numbers are multiplied, each number that is multiplied is called a factor. For example, when we write 3 × 7 × 5, each of the numbers 3, 7 and 5 is called a factor.

Prime numbers are all natural numbers greater than 1 whose only natural number factors are 1 and itself. The number 5 is prime. The only natural number factors of 5 are 5 and 1.

5 = 5 × 1

The number 6 is not prime. The natural number factors of 6 are 3 and 2 or 6 and 1.

6 = 3 × 2 6 = 6 × 1

The first 15 prime numbers are

2, 3, 9, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Any natural number greater than 1 is either prime or can be written as the product of prime numbers. For example we can take each of the numbers 12, 30, 14, 19 and 29 and either indicate that they are prime or, if they are not prime, write them as the product of prime numbers. We write as follows:

12 = 2 × 2 × 3 30 = 2 × 3 × 5 14 = 2 × 7

19 is a prime number. 29 is a prime number.

To reduce a fraction, we use prime numbers to factor numerator and denominator. Write each part of the fraction (numerator and denominator) as a product of prime numbers. Note any factor that appears in both the numerator (top) and denominator (bottom) of the fraction. If we divide numerator and denominator by that value, we will obtain an equivalent fraction in simplest form. When the new fraction is simplified, it is said to be in lowest terms. Throughout this text, when we say simplify a fraction, we always mean to lowest terms.

EXAMPLE 1 Simplify each fraction. (a) 14/21 (b) 15/35 (c) 20/70

(a) 14/21=(7*2)/(7*3)=2/3

We factor 14 and factor 21. Then we divide numerator and denominator by 7.

(b) 15/35=(5*3)/(5*7)=3/7

We factor 15 and factor 35. Then we divide numerator and denominator by 5.

We factor 20 and factor 70. Then we divide numerator and denominator by both 2 and 5.

Sometimes when we simplify a fraction, all the prime factors in the top (numerator) are divided out. When this happens, we put a 1 in the numerator. If we did not put the 1 in the numerator, we would not realize that the answer was a fraction.

EXAMPLE 2 Simplify each fraction. (a) 7/21 (b) 15/105

(a) 7/21=(7*1)/(7*3)=1/3 (b) 15/105=(5*3*1)/(7*5*3)=1/7

If all the prime numbers in the bottom (denominator) are divided out, we do not need to leave a 1 in the denominator, since we do not need to express the answer as a fraction. The answer is then a whole number and is not usually expressed as a fraction.

EXAMPLE 3 Simplify each fraction. (a) 35/7 (b) 70/10

(a) 35/7=(5*7)/7=5 (b) 70/10=(7*5*2)/(5*2)=7

Sometimes the fraction we use represents how many of a certain thing are successful. For example, if a major league baseball player was at bat 30 times and achieved 12 hits, we could say that he had a hit 12/30 of the time. If we reduce the fraction, we could say he had a hit 2/5 of the time.

EXAMPLE 4 Cindy got 48 out of 56 questions correct on a test. Write this as a fraction.

Express as a fraction in simplest form the number of correct responses out of the total number of questions on the test.

48 out of 56 ⟶ 48/56=(6*8)/(7*8)=6/7

Cindy answered the questions correctly 6/7 of the time.

The number one can be expressed as 1, 1/1, 2/2, 6/6, 8/8 and so on, since

1= 1/1= 2/2= 6/6= 8/8

We say that these numerals are equivalent ways of writing the number one because they all express the same quantity even though they appear to be different.

SIDELIGHT When we simplify fractions, we are actually using the fact that we can multiply any number by 1 without changing the value of that number. (Mathematicians call the number 1 the multiplicative identity because it leaves any number it multiplies with the same identical value as before.)

Let’s look again at one of the previous problems.

14/21 = (7*2)/(7*3) = 7/7 × 2/3 = 1 × 2/3

So we see that
14/21=2/3

When we simplify fractions, we are using this property of multiplying by 1.

Improper Fractions and Mixed Numbers

If the numerator is less than the denominator the fraction is a proper fraction. A proper fraction is used to describe a quantity smaller than a whole.

Fractions can also be used to describe quantities larger than a whole. The figure below shows two bars that are equal in size. Each bar is divided into 5 equal pieces. The first bar is shaded in completely. The second bar has 2 of the 5 pieces shaded in.

The shaded-in region can be represented by since 7 of the pieces (each of which is 5 of a whole box) are shaded. The fraction 7/5 is called an improper fraction. An improper fraction is one in which the numerator is larger than or equal to the denominator.

The shaded-in region can also be represented by 1 whole added to 2/5 of a whole, or 1 + 2/5. This is written as 12/5. The fraction 12/5 is called a mixed number. A mixed number consists of a whole number added to a proper fraction (numerator is smaller than the denominator). The addition is understood but not written. When we write 12/5, it represents 1 + 2/5. The numbers 17/8, 23/4, 81/3, and 1261/10 are all mixed numbers. From the figure above it seems clear that 7/5 = 12/5. This suggests that we can change from one form to the other without changing the value of the fraction.

From a picture it is easy to see how to change improper fractions to mixed numbers. For example, if we start with the fraction 11/3 and represent it by the figure below (where 11 of the pieces that are 1/3 of a box are shaded), we see that 11/3 = 32/3, since 3 whole boxes and & 2/3 of a box are shaded.

Changing Improper Fractions to Mixed Numbers

You can do the same procedure without a picture. For example, to change 11/3 to a mixed number, we can do the following:

11/3=3/3+3/3+3/3+2/3 By the rule for adding fractions (which is discussed in detail in next section).

= 1+1+1+2/3 Write 1 in place of 3/3, since 3/3 = 1.

= 3+ 2/3 Write 3 in place of 1+ 1+ 1.

= 32/3 Use the notation for mixed numbers.

Now that you know how to perform the change and why it works, here is a shorter method.

To Change an Improper Fraction to a Mixed Number
| 1. Divide the denominator into the numerator.
2. The result is the whole-number part of the mixed number.
3. The remainder from the division will be the numerator of the fraction. The
denominator of the fraction remains unchanged.

We can write the fraction as a division example and divide. The arrows show how to write the mixed number.

7 1 Whole-number part > Numerator of fraction
==5)7 I=
5 5 5
2 Remainder
7 2
Thus, — = 1-.
us 5 5
2 ;
3 Whole-number part ——> 3— | Numerator of fraction
11 3
—=3)11
3 9
2 Remainder
ll 2
Thus, — = 3—-.
m3 3

Sometimes the remainder is 0. The improper fraction changes to a whole number.

EXAMPLE 5 Change to a mixed number or to a whole number.

(a) 7/4 (b) 15/3

(a) 4 (b) 3
7 15 >
(a) —=7+4=4)7 (b) — = 15+3 = 3)15
4 3
4 IS
3 Remainder Q Remainder
7 3 15
—=]|1-. Thus — = 5.

Thus 7/4 = 13/4 Thus 15/3 = 5

Changing Mixed Numbers to Improper Fractions

It is not difficult to see how to change mixed numbers to improper fractions. Suppose that you wanted to write 22/3 as an improper fraction:

22/3 = 2+2/3 The meaning of mixed number notation.

= 1+1+2/3 Since 1+ 1 = 2,

= 3/3+3/3+2/3 Since 1 = 3/3.

When we draw a picture of 3/3+3/3+2/3, we have this figure:

3 3 2
3 3 3

If we count the shaded parts, we see that

3/3+3/3+2/3=8/3 Thus 22/3 = 8/3

Now that you can see how this change can be done, here is a shorter method.

To Change a Mixed Number to an Improper Fraction
1. Multiply the whole number by the denominator.
2. Add this to the numerator. The result is the new numerator. The denominator
does not change.
. . l 4

EXAMPLE 6 Change to an improper fraction. (a) 31/7 (b) 54/5

(a) 31/7 = ((3*7)+1)/7=(21+1)/7=22/7 (b) 54/5 = ((5*5)+4)/5=(25+4)/5=29/5

Changing a Fraction to an Equivalent Fraction with a Different Denominator

Fractions can be changed to an equivalent fraction with a different denominator by multiplying both numerator and denominator by the same number.

5/6=(5*2)/(6*2)=10/12 3/7=(3*3)/(7*3)=9/21

So 5/6 is equivalent to 10/12. 3/7 is equivalent to 9/21.

We often want to get a particular denominator.

EXAMPLE 7 Find the missing number.

(a)
@) s 25 (0) 2 21 > 36
3 ?
(a) 5 = 55 Observe that we need to multiply the denominator by 5 to
“ obtain 25. So we multiply the numerator 3 by § also.
3x5 15
—— = = The desired numerator is 15.
3X5 25
3 ?
(b) a = a1 Observe that 7 X 3 = 21. We need to multiply by 3 to get the
new numerator.
3% 3 ? The desired tor is 9
—_—_- = — 1¢@ desired numerator is 9.
7X3 21 : mime
2 ?
(c) 9 = 36 Observe that 9 X 4 = 36. We need to multiply by 4 to get the
" new numerator.
ext The desired tor is 8
9x4 36 1e desired numerator is 8.

Addition and Subtraction of Fractions

After studying this section, you will be able to:

1. Add or subtract fractions with a common denominator.

2. Find the least common denominator of any two or more fractions.

3. Add or subtract fractions that do not have a common denominator.

4. Add or subtract mixed numbers.

Adding or Subtracting Fractions with a Common Denominator

If fractions have the same denominator, the numerators may be added or subtracted. The denominator remains the same.

To Add or Subtract Two Fractions with the Same Denominator
1. Add or subtract the numerators.
2. Keep the same denominator.
3. Simplify the answer whenever possible.

EXAMPLE 1 Add the fractions. Simplify your answer whenever possible.

(a) 5/7+1/7 (b) 2/3+1/3 (c) 1/8+3/8+2/8 (d) 3/5+4/5

(a) 5/7+1/7=(5+1)/7=6/7 (b) 2/3+1/3=(2+1)/3=3/3=1

EXAMPLE 2 Subtract the fractions. Simplify your answer whenever possible.

(a) 9/11-2/11 (b) 5/6-1/6

(a) 9/11-2/11=(9-2)/11=7/11 (b) 5/6-1/6=(5-1)/6=4/6=2/3

Our ability to add and subtract fractions with a common denominator allows us to solve simple problems. Most problems, however, involve fractions that do not have a common denominator. Fractions and mixed numbers such as halves, fourths, and eighths are commonly used. To add or subtract such fractions, we begin by finding a common denominator.

Finding the Least Common Denominator

Before you can add or subtract, fractions must have the same denominator. To save work, we select the smallest possible common denominator. This is called the least common denominator or LCD (also known as the lowest common denominator ).

The LCD of two or more fractions is the smallest whole number that is exactly divisible by each denominator of the fractions.

EXAMPLE 3 Find the LCD. 2/3 and 1/4

The numbers are small enough to find the LCD by inspection. The LCD is 12, since 12 is exactly divisible by 4 and by 3. There is no smaller number that is exactly divisible by 4 and 3.

In some cases, the LCD cannot easily be determined by inspection. If we write each denominator as the product of prime factors, we will be able to find the LCD. We will use (.) to indicate multiplication. For example, 30 = 2*3*5. This means 30 = 2 × 3 × 5.

Procedure to Find the LCD Using Prime Factors

1. Write each denominator as the product of prime factors.

2. The LCD is the product of each different factor.

3. If a factor occurs more than once in any one denominator, the LCD will contain
that factor repeated the greatest number of times that it occurs in any one denom-
inator.

EXAMPLE 4 Find the LCD of 5/6 and 1/15 by this new procedure.

6=2:-3 Write each denominator as the product of prime factors.
IS = | 3°5
Vy
LCD =2:-3:5
LCD = 2-3-5 = 30 The LCD is the product of each different prime factor. The
different factors are 2, 3, and 5.
Practice Problem 4 Use the prime factors to find the LCD of & and &. m=

Great care should be used to determine the LCD in the case of repeated factors.
EXAMPLE 5 Find the LCD of 34 and 34.
27=3:-3°-3 Write each denominator as the product of prime factors.
We observe that the factor 3 occurs three times in the
1I8=] 3:3:-2 or
| | | factorization of 27.
LCD =3-3-3-2
LCD =3-3°-3-:2=54
The LCD is the product of each different factor. The factor 3 occurred most in the
factorization of 27, where it occurred three times. Thus the LCD will be the product of
three 3’s and one 2.
Practice Problem 5 Find the LCD of 35 and 3%. a
EXAMPLE 6 Find the LCD of #5, 7s, and 4.
12=2-2-3 Write each denominator as the product of prime factors.
Notice that the repeated factor is 2, which occurs twice
15 = 3-5 ; ge
in the factorization of 12.
: / | | |
LCD =2-2-3-5
LCD =2-2-3-5=60
The LCD its the product of each different factor with the factor 2 repeated since it occurred
twice in one denominator.
Practice Problem 6 Find the LCD of #, 3k. and 35. =

Adding or Subtracting Fractions That Do Not Have a Common Denominator

Before you can add or subtract them, fractions must have the same denominator. Using the LCD will make your work easier. First you must find the LCD. Then change each fraction to a fraction that has the LCD as the denominator. Sometimes one of the fractions will already have the LCD as the denominator. Once all the fractions have the same denominator, you can add or subtract. Be sure to simplify the fraction in your answer if this is possible.

To Add or Subtract Fractions That Do Not Have a Common
Denominator
1. Find the LCD of the fractions.
2. Change each fraction to an equivalent fraction with the LCD for a denominator.
3. Add or subtract the fractions.
4. Simplify the answer whenever possible.

Let us return to the two fractions of Example 3. We have previously found that the LCD is 12.

EXAMPLE 7 Add. 3 + 4
We must change § and } to fractions with the same denominator. We change each
fraction to an equivalent fraction with a common denominator of 12, the LCD.
2 ? 2 4 8 2 8
_—-_ = — XK —_>_ = SO —- =
3 12 3 4 #12 3 12
l ? 1 3 3 ] 3
7 =a —x—-=—_ so -=—
4 12 4 3 12 4 12
Then we rewrite the problem with common denominators and then add.
2 | 8 3 8+3 11
3 4 12 = «12 12 12
; ] 5
Practice Problem 7 Add. 3 D ®
Sometimes one of the denominators is the LCD. In such cases the fraction that has
the LCD for the denominator will not need to be changed. If every other denominator
divides into the largest denominator, the largest denominator is the LCD.
3 7 l
EXAMPLE 8 Find the LCD and then add. 5 + 30 + 5
We can see by inspection that both 5 and 2 divide exactly into 20. Thus 20 is the
LCD. Now add:
3 7 l
5 20 2
We change # and $ to equivalent fractions with a common denominator of 20, the LCD.
3 ? 3 4 #12 3 «12
_—_ = —_—_—xK_-_ =a SO _—_- =>
5 20 5 4 20 5 20
l ? 1 10 #10 1 10
-— => —— el sooo =
2 20 2 10 20 2 20
Then we rewrite the problem with common denominators and add:
3 7 1 12 7 10 12+7+10 29 9
zt tmt+r e+ H+ 8 = = Oo“ 1S
5 20 2 20 20 ~~ 20 20 20 20
, . 3 4 l
Practice Problem 8 Find the LCD and add. 3 + 357 70"

Now we turn to examples where the selection of the LCD is not so obvious. In
Examples 9 through 11 we will use the prime factorization method to find the LCD.
EXAMPLE 9 Add. —— + —

18 12
First we find the LCD.
1I8=3-3-2
12= | 3°2:2
LCD =3°3°2°2=9:4= 36
Now we change 7% and 35 to equivalent fractions that have the LCD.
Tot Fy ye lt
18 36 18 2 36
Sof Sy3Lb
12 36 12 3 36
Now we add the fractions.
- + o = _ + = = = This fraction cannot be simplified.
Practice Problem 9 Add. 4 + 3 =
49 14
EXAMPLE 10. Subtract. — — —
48 36
First we find the LCD.
48 =2°-2-2-2-3
36 = | | 2:2:°3:-3
LCD =2:2-°:2-2-3-3=16°9= 144
Now we change 4% and 3 to equivalent fractions that have the LCD for a denominator.
2 2838 1S
48 144 48 3 144
St 8 yA 20
36 =—s «144 36 6©4—=«(«144
Now we subtract the fractions.
> = Po = >> This fraction cannot be simplified.
48 36 144 144 144
Practice Problem 10 Subtract. on "
12 =—30

EXAMPLE 11 Add. — + — + —
5 6 10
First we find the LCD.
5=5
6= | 2°3
y
10=5:2
LCD = 5:2:3 = 10:3 = 30
Now we change 4, 4, and 35 to equivalent fractions that have the LCD for a denominator.
1? 1,6 6
5 30 5 6 30
1 ? 1 § 5
—_ = _ — X — = ——_
6 30 6 5 30
3B? 3,39
10 30 10 3 #30
Now we add the three fractions.
1] l 3 6 5 9 20 2
+> 4+ —=—4+—4+—=— ==
5 6 10 30 30 30 30 3
Note the important step of simplifying the fraction to obtain the final answer.
, 2 3 3
Practice Problem 11 Combine. 3 + 473 a

Adding or Subtracting Mixed Numbers

If the problem you are adding or subtracting has mixed numbers, change them to improper fractions first and then combine (addition or subtraction). Often the final answer is changed to a mixed number. Here is a good rule to follow:

If the original problem contains mixed numbers, express the result as a mixed
number rather than as an improper fraction.

EXAMPLE 12 Combine. Simplify your answer whenever possible.

(a) 51/2 + 21/3 (b) 21/5 - 13/4 (c) 15/12 + 7/30

(a) First we change the mixed numbers to improper fractions.

51/2 = (5*2+1)/2=11/2 21/3 = (2*3+1)/3=7/3

Next we change each fraction to an equivalent form with a common denominator of 6.

Mo? U3 3
2 6 2 3 6
7 ? 7 2 14
— —-x-=-—
3 «66 3 2 6

Finally, we add the two fractions and change our answer to a mixed number.

33/6+14/6=47/6 = 75/6

Thus, 51/2 + 21/3 = 75/6

(b) First we change mixed numbers to improper fractions.

21/5 = (2*5+1)/5=11/5 13/4 = (1*4+3)/4=7/4

Next we change each fraction to an equivalent form with a common denominator of 20.

nl? M4 M4
5 20 5 4 20
7TH? 7,3»
4 20 4 5 20

Now we subtract the two fractions.

44/20-35/20=9/20

Thus 21/5 - 13/4 = 9/20.

Note: It is not necessary to use these exact steps to add and subtract mixed numbers. If you know another method and can use it to obtain the correct answers, it is all right to continue to use that method throughout this chapter.

(c) Now we add 15/12 + 7/30

The LCD of 12 and 30 is 60. Why? Change the mixed number to an improper fraction. Then change each fraction to an equivalent form with a common denominator.

15/12 = 17/12 × 5/5 = 85/60 7/30 × 2/2 = 14/60

Then add the fractions, simplify, and write the answer as a mixed number.

85/60+14/60=99/60=33/20 = 113/20

Thus, 15/12 + 7/30 = 113/20

EXAMPLE 13 Manuel is enclosing a triangular-shaped exercise yard for his new dog. He wants to determine how much fencing he will need. The sides of the yard measure 203/4 feet, 151/2 feet, and 181/8 feet. What is the perimeter of (total distance around) the triangle?

Understand the problem. Begin by drawing a picture.
3
207 ft 155 ft
185 ft

We want to add up the lengths of all three sides of the triangle. This distance around the triangle is called the perimeter.

203/4 + 151/2 + 181/8 = 83/4+31/2+145/8

= 166/8+124/8+145/8=435/8 = 543/8 feet.

Multiplication and Division of Fractions

After studying this section, you will be able to:

1. Multiply fractions, whole numbers, and mixed numbers.

2. Divide fractions, whole numbers, and mixed numbers.

Multiplying Fractions, Whole Numbers, and Mixed Numbers

Multiplication of Fractions

The multiplication rule for fractions states that to multiply two fractions we multiply the two numerators and multiply the two denominators.

To Multiply Any Two Fractions
1. Multiply the numerators.
2. Multiply the denominators.

Consider these examples.

EXAMPLE 1 Multiply.

(a) 3/5 × 2/7 (b) 1/3 × 5/4 (c) 7/3 × 1/5 (d) 6/5 × 2/3

(a) 3/5 × 2/7=(3*2)/(5*7)=6/35 (b) 1/3 × 5/4=(1*5)/(3*4)=5/12

It is possible to avoid having to simplify a fraction at the last step. In many cases we can divide by a value that appears as a factor in both a numerator and a denominator. Often it is helpful to write a number (as a product of prime factors) in order to do this.

EXAMPLE 2 Multiply.

(a) 3/5 × 5/7 (b) 4/11 × 5/2 (c) 15/8 × 10/27 (d) 8/7 × 5/12

(a) 3/5 × 5/7 = (3*5)/(5*7)=1/7 Note that here we divided numerator and denominator by 5.

If we factor each number, we can see the common factors.

(b) 4/11 × 5/2 = (2*2)/11 × 5/2 = 10/11

(d) 8/7 × 5/12 = (2*2*2)/7 × 5/(2*2*3) = 10/21

By dividing out common factors, the multiplication involves smaller numbers and the answers are in simplified form.

SIDELIGHT Why does this method of dividing out a value that appears as a factor in both numerator and denominator work? Let’s reexamine one of the examples we have solved previously.

3/5 × 5/7

Consider the following steps and reasons.

3/5 × 5/7 = (3*5)/(5*7) Definition of multiplication of fractions.

= (5*3)/(5*7) Change the order of the factors in the numerator, since 3*5 =5*3. This is called the commutative property of multiplication.

= 5/5*3/7 Definition of multiplication of fractions.

= 1*3/7 Write 1 in place of 5/5 since 1 is another name for 5/5.

= 3/7 1*3/7=3/7 since any number can be multiplied by 1 without changing the value of the number.

Think about this concept. It is an important one that we will use again when we discuss rational expressions.

Multiplication of a Fraction by a Whole Number

Whole numbers can be named using fractional notation. 3, 9/3, 6/2 and 3/1 are ways of expressing the number three. Therefore,

3=9/3=6/2=3/1

When we multiply a fraction by a whole number, we merely express the whole number as a fraction whose denominator is 1 and follow the multiplication rule for fractions.

EXAMPLE 3 Multiply. (a) 7 × 3/5 (b) 3/16 × 4

(a) 7 × 3/5 = 7/1 × 3/5 = 21/5 = 41/5 (b) 3/16 × 4 = 3/16 × 4/1 = 3/(4*4) × 4/1 = 3/4

Notice that in (b) we did not use prime factors to factor 16. We recognized that 16 = 4*4. This is a more convenient factorization of 16 for this problem. Choose the factorization that works best for each problem. If you cannot decide what is best, factor into primes.

Multiplication of Mixed Numbers

When multiplying mixed numbers, we first change them to improper fractions and then follow the multiplication rule for fractions.

EXAMPLE 4 Multiply. (a) 31/3 × 21/2 (b) 12/5 × 21/3

(a) 31/3 × 21/2 = 10/3 × 5/2 = (2*5)/3 × 5/2 = 25/3 = 81/3

(b) 12/5 × 21/3 = 7/5 × 7/3 = 49/15 = 34/15

Dividing Fractions, Whole Numbers, and Mixed Numbers

Division of Fractions

To divide two fractions, we invert the second fraction and then multiply the two fractions.

To Divide Two Fractions
1. Invert the second fraction.
2. Now multiply the two fractions.

EXAMPLE 5 Divide. (a) 1/3 ÷ 1/2 (b) 2/5 ÷ 3/10 (c) 2/3 ÷ 7/5

(a) 1/3 ÷ 1/2 = 1/3 × 2/1 = 2/3

(b) 2/5 ÷ 3/10 = 2/5 × 10/3 = 2/5 × (5*2)/3 = 4/3 = 11/3

Division of a Fraction and a Whole Number

The process of inverting the second fraction and then multiplying the two fractions should be done very carefully when one of the original values is a whole number. Remember, a whole number such as 2 is equivalent to 2/1.

EXAMPLE 6 Divide. (a) 1/3 ÷ 2 (b) 5 ÷ 1/3

(a) 1/3 ÷ 2 = 1/3 ÷ 2/1 = 1/3 × 1/2 = 1/6

(b) 5 ÷ 1/3 = 5/1 ÷ 1/3 = 5/1 × 3/1 = 15/1 = 15

SIDELIGHT Number Sense Look at the answers to the problems in Example 6. In part (a), you will notice that % is less than the original number 4. Does this seem reasonable? Let’s see. If 1/3 is divided by 2, it means that 1/3 will be divided into 2 equal parts. We would expect that each part would be less than 1/3. 1/6 is a reasonable answer to this division problem.

In part (b), 15 is greater than the original number 5. Does this seem reasonable? Think of what 5 ÷ 1/3 means. It means that 5 will be divided into thirds. Let’s think of an easier problem. What happens when we divide 1 into thirds? We get three thirds. We would expect, therefore, that when we divide 5 by thirds, we would get 5 × 3 or 15 thirds. 15 is a reasonable answer to this division problem.

Complex Fraction

Sometimes division is written in the form of a complex fraction with one fraction in the numerator and one fraction in the denominator. It is best to write this in standard division notation first; then complete the problem using the rule for division.

EXAMPLE 7 Divide. (a) (3/7)/(3/5) (b) (2/9)/(5/7)

(a) (3/7)/(3/5) = 3/7 ÷ 3/5 = 3/7 × 5/3 = 5/7 (b) (2/9)/(5/7) = 2/9 ÷ 5/7 = 2/9 × 7/5 = 14/45

SIDELIGHT Why does the method of ‘‘invert and multiply’’ work? The division rule really depends on the property that any number can be multiplied by 1 without changing the value of the number. Let’s look carefully at an example of division of fractions:

2/5 ÷ 3/7 = (2/5)/(3/7) We can write the problem using a complex fraction.

= (2/5)/(3/7) × 1 We can multiply by 1, since any number can be multiplied by 1 without changing the value of the number.

= (2/5)/(3/7) × (7/3)/(7/3) We write 1 in the form (7/3)/(7/3) since any nonzero number divided by itself equals 1. We choose this value as a multiplier because it will help simplify the denominator.

= (2/5*7/3)/(3/7*7/3) Definition of multiplication of fractions.

= (2/5*7/3)/1 The product in the denominator equals 1.

= 2/5 × 7/3

Thus we have shown that 2/5 ÷ 3/7 is equivalent to 2/5 × 7/3 and have shown some justification for the ‘‘invert and multiply rule.”

Division of Mixed Numbers

This method for division of fractions can be used with mixed numbers. However, we first must change the mixed numbers to improper fractions and then use the rule for dividing fractions.

EXAMPLE 8 Divide. (a) 21/3 ÷ 32/3 (b) 41/2 ÷ 15/7

(a) 21/3 ÷ 32/3 = 7/3 ÷ 11/3 = 7/3 × 3/11 = 7/11

(b) 41/2 ÷ 15/7 = 9/2 ÷ 12/7 = 9/2 × 7/12 = (3*3)/2 × 7/(4*3) = 21/8 = 25/8

EXAMPLE 9 A chemist has 96 fluid ounces of a solution. She pours the solution into test tubes. Each test tube holds 3/4 fluid ounce. How many test tubes can she fill?

We need to divide the total number of ounces, 96, by the number of ounces in each test tube, 3/4.

96 ÷ 3/4 = 96/1 ÷ 3/4 = 96/1 × 4/3 = (3*32)/1 × 4/3 = 128/1 = 128

She will be able to fill 128 test tubes.

Pause for a moment to think about the answer. Does 128 test tubes filled with solution seem like a reasonable answer? Did you perform the correct operation?

Sometimes when solving word problems involving fractions, or mixed numbers, it is helpful to solve the problem using simpler numbers. Once you understand what operation is involved, you can go back and solve using the original numbers in the word problem.

Use of Decimals

After studying this section, you will be able to:

1. Change from a fraction to a decimal.

2. Change from a decimal to a fraction.

3. Add and subtract decimals.

4. Multiply decimals.

5. Divide decimals.

6. Multiply or divide a number by a multiple of 10.

The Basic Concept of Decimals

We can express a part of a whole as a fraction or as a decimal. A decimal is another way of writing a fraction whose denominator is 10, 100, 1000, and so on.

3/10=0.3 5/100=0.05 172/1000=0.172 58/10000=0.0058

The period in decimal notation is known as the decimal point. The number of digits in a number to the right of the decimal point is known as the number of decimal places of the number. The place value of decimals are shown below.

EXAMPLE 1 Write the following decimals as a fraction. Give the number of decimal
places. Write out in words the way the number would be spoken.

(a) 0.6 (b) 0.29 (c) 0.527

(d) 1.38 (e) 0.00007

You have seen that a given fraction can be written in several different but equivalent ways. There are also several different equivalent ways of writing the decimal form of fractions. The decimal 0.18 can be written in the following equivalent ways:

Fractional form: 18/100=180/1000= 1800/10000= 18000/100000

Decimal form: 0.18 = 0.180 = 0.1800 = 0.18000

Thus we see that any number of terminal zeros may be added onto the right-hand side of a decimal without changing its value.

0.13 = 0.1300 0.162 = 0.162000

Similarly, any number of terminal zeros may be removed from the right-hand side of a decimal without changing its value.

Changing a Fraction to a Decimal

A fraction can be changed to a decimal by dividing the denominator into the numerator.

EXAMPLE 2 Write each of the following fractions as a decimal.

(a) 3/4 (b) 21/20 (c) 1/8 (d) 3/200

Sometimes division yields an infinite repeating decimal. We use three dots to indicate that the pattern continues forever. For example:

An alternative notation is to place a bar over the repeating digits:

0.3333... = 0.3 0.575757... = 0.57

EXAMPLE 3 Write each fraction as a decimal. (a) 2/11 (b) 5/6

Sometimes division must be carried out to many places in order to observe the repeating pattern. This is true in the following example:

2/7=0.285714285714285714 ... This can also be written as 2/7 = 0.285714.

It can be shown that the denominator determines the maximum number of decimal places that might repeat. So 2/7 must repeat in the seventh decimal place or sooner.

Changing a Decimal to a Simple Fraction

To convert from a decimal to a fraction, merely write the decimal as a fraction with a denominator of 10, 100, 1000, 10,000 and so on, and simplify the result when possible.

EXAMPLE 4 Write each decimal as a fraction.

(a) 0.2 (b) 0.35 (c) 0.516 (d) 0.74

(a) 0.2=2/10=1/5 (b) 0.35=35/100=7/20

All repeating decimals can also be converted to fractional form. In practice, however, repeating decimals are usually rounded to a few places. It will not be necessary, therefore, to learn how to convert 0.033 to 11/333 for this course.

Adding and Subtracting Decimals

Adding or subtracting decimals is similar to adding and subtracting whole numbers except that it is necessary to line up decimal points. To perform the operation 19.8 + 24.7 we line up the numbers in column form and add the digits
19.8
+ 24,7
44.5

Addition and Subtraction of Decimals
1. Write in column form and line up decimal points.
2. Add or subtract the digits.

EXAMPLE 5 Perform the following operations.

(a) 3.6 + 2.3 (b) 127.32 - 38.48

(a) 36 £(b) = 127.32 = (e) 3.1 (d) 5.0006
+ 2.3 — 38.48 42.36 — 3.1248
5.9 88.84 + 9.034 1.8758
54.494

SIDELIGHT When we added fractions, we had to have common denominators. Since decimals are really fractions, why can we add them without having common denominators? Actually, we have to have common denominators to add any fractions, whether they are in decimal form or fraction form. However, sometimes the notation does not show this.

Let’s examine Example 5(c), worked above.
Original Problem We are adding the three numbers:
3.1 3¢5 + 42705 + 97860
42.56 37800 + 42z000 + Srbd5
+ 7.034 3.100 + 42.360 + 9.034 This is the New Problem.
54.494
New Problem Original Problem
3.100 3.1 We notice that the results are the same. The
42.360 42.36 only difference is the notation. We are using
+ 9.034 + 9.034 the property that any number of zeros may
54.494 54.494 be added to the right-hand side of a
decimal without changing its value.

This shows the convenience of adding and subtracting fractions in the decimal form. Little work is needed to change the decimals to a common denominator. All that is required is to add zeros to the right-hand side of the decimal (and we usually do not even write out that step except when subtracting).

As long as we line up the decimal points, we can add or subtract any decimal fractions.

In the following example we will find it useful to add zeros to the right-hand side of the decimal.

EXAMPLE 6 Perform the following operations.

(a) 1.0003 + 0.02 + 3.4 (b) 12 - 0.057

We will add zeros so that each number shows the same number of decimal places.
(a) 1.0003 (b) 12.000
0.0200 — 0.057
+ 3.4000 11.943
4.4203

Practice Problem 6 Perform the following operations.
(a) 0.061 + 5.0008 +13 (b) 18-—0.126 »

Multiplying Decimals

Multiplication of Decimals
To multiply decimals, you first multiply as with whole numbers. To determine the
position of the decimal point, you count the total number of decimal places in the
two numbers being multiplied. This will determine the number of decimal places
that should appear in the answer.

EXAMPLE 7 Multiply 0.8 × 0.4

0.8 (one decimal place)
x 0.4 (one decimal place)
0.32 (two decimal places)

Note that you will often have to add zeros to the left of the digits obtained in the product so that you obtain the necessary number of decimal places.

EXAMPLE 8 Multiply 0.123 × 0.5

0.123 (three decimal places)
x 0.5 (one decimal place)
0.0615 (four decimal places)
Practice Problem 8 Multiply. 0.12 x 0.4 s

Here are some examples that involve more decimal places.

EXAMPLE 9 Multiply (a) 2.56 × 0.003 (b) 0.0036 x 0.008

(a) 2.56 (two decimal places) (b) 0.0036 (four decimal places)
Xx 0.003 (three decimal places) X 0.008 (three decimal places)
0.00768 (five decimal places) 0.0000288 (seven decimal places)

SIDELIGHT Why do we count the number of decimal places? The rule really comes from the properties of fractions. If we write the problem in Example 8 in fraction form, we have
123 5 615
0.123) X (0.5) = —— x — = —— = 0.0615
( ) x (0.5) 1000 =10 ~= 10,000

Dividing Decimals

When discussing division of decimals, we frequently refer to the three primary parts of a division problem. Be sure you know the meaning of each term.

The divisor is the number you divide into another.

The dividend is the number to be divided.

The quotient is the result of dividing one number by another.

In the problem 6 ÷ 2 = 3 we represent each of these terms as follows:

3<— Quotient Quotient
Divisor ——> a6 Divisor)Dividend
Dividend

When dividing two decimals, count the number of decimal places in the divisor.
Then move the decimal point to the right that same number of places in both the
divisor and the dividend. Mark that position with a caret (,). Finally, perform the
division. Be sure to line up the decimal point in the quotient with the position
indicated by the caret in the dividend.

EXAMPLE 10 Divide 32.68 ÷ 4

8.17 Since there are no decimal places in the divisor, we do not
4)32.68 need to move the decimal point. We must be careful, however, to
32 place the decimal point in the quotient directly above the
6 decimal point in the dividend.
4
28
28

Thus 32.68 ÷ 4 = 8.17.

EXAMPLE 11 Divide 5.75 ÷ 0.5

0.5 )5.7. 5 There is one decimal place in the divisor, so we move the
One decimal place decimal point one place to the right in the divisor and
P dividend and we mark that new position by a caret (,).
11.5
0.5 )5.7, 5 Now we perform the division as with whole numbers. The
5 decimal point in the answer is directly above the caret in
47 the dividend.
>
25
25

Thus 5.75 ÷ 0.5 = 11.5

Note that sometimes we will need to place extra zeros in the dividend in order to move the decimal point the required number of places.

EXAMPLE 12 Divide 16.2 ÷ 0.027

0,027, )16,200 There are three decimal places in the divisor, so we move
the decimal point three places to the right in the divisor
and dividend and mark the new position by a caret. Note
Three decimal places = that we must add two zeros to 16.2 in order to do this.
600.
0,027 )16.200 Now perform the division as with whole numbers. The
163 decimal point in the answer is directly above the caret in
000 the dividend.

Thus 16.2 ÷ 0.027 = 600.

Special care must be taken to line up the digits in the quotient. Note that sometimes we will need to place zeros in the quotient after the decimal point.

EXAMPLE 13 Divide 0.04288 ÷ 3.2

3.2 )0,0 4288 There is one decimal place in the divisor, so we move the
decimal point one place to the right in the divisor and
dividend and mark the new position by a caret.
One decimal place
0.0134
3.2 )0.0 4288 Now perform the division as for whole numbers. The decimal
32 point in the answer is directly above the caret in the dividend.
108 Note the need for the initial zero after the decimal point in
96 the answer.
128
128
0

Thus 0.04288 ÷ 3.2 = 0.0134.

SIDELIGHT Why does this method of dividing decimals work? Essentially. we are using the steps we used in Section 0.1 to change a fraction to an equivalent fraction by multiplying both the numerator and denominator by the same number. Let’s reexamine Example

13.
0.04288
0.04288 + 3.2 = 39, Write the original problem using fraction notation.
0.04288 x 10 Multiply the numerator and denominator by
= 32 x10 10. Since this is the same as multiplying by 1,
. we are not changing the fraction.
0.4288
= 39 Write the result of multiplication by 10.
= 0.4288 + 32 Rewrite the fraction as an equivalent problem
with division notation.

Notice that we have obtained a new problem that is the same as the problem in Example 13 when we moved the decimal one place to the right in the divisor and dividend. We see that the reason we can move the decimal point so many places to the right in divisor and dividend is that the numerator and denominator of a fraction are both being multiplied by 10, 100, 1000 and so on, to obtain an equivalent fraction.

Multiplying and Dividing by a Multiple of 10

When multiplying by 10, 100, 1000 and so on, a simple rule may be used to obtain the answer. For every zero in the multiplier, move the decimal point one place to the right.

EXAMPLE 14 Multiply (a) 3.24 × 10 (b) 15.6 × 100 (c) 0.0026 × 1000

(a) 3.24 × 10 = 32.4 One zero—move decimal point one place to the right.

(b) 15.6 × 100 = 1560 Two zeros—move decimal point two places to the right.

(c) 0.0026 × 1000 = 2.6 Three zeros—move decimal point three places to the right.

The reverse rule is true for division. When dividing by 10, 100, 1000, 10,000 and so on, move the decimal point one place to the left for every zero in the divisor.

EXAMPLE 15 Divide (a) 52.6 ÷ 10 (b) 0.0038 ÷ 100 (c) 5936.2 ÷ 1000

(a) 52.6/10=5.26 Move one place to the left.

(b) 0.0038/100=0.000038 Move two places to the left.

(c) 5936.2/1000=5.9362 Move three places to the left.

Use of Percent

After studying this section, you will be able to:

1. Change a decimal to a percent.

2. Change a percent to a decimal.

3. Find the percent of a number.

4. Find the missing percent when given two numbers.

The Basic Concept of Percents

A percent is a fraction that has a denominator of 100. When you say ‘‘sixty-seven percent’’ or write 67%, you are just using another way of expressing the fraction 67/100. The word percent is a shortened form of the Latin words per centum, which means ‘‘by the hundred.” In everyday use, percent means per one hundred.

It is important to see that 49% means 49 parts out of 100 parts. It can also be written as a fraction, 49/100, or as a decimal: 0.49. Understanding the meaning of the notation allows you to change from one notation to another. For example,

49% = 49 out of 100 parts = 49/100 = 0.49

Similarly, you can express a fraction with denominator 100 as a percent or a decimal.

11/100 means 11 parts out of 100 or 11%; 11/100 as a decimal is 0.11

So 11/100 = 11% = 0.11

Changing a Decimal to a Percent

Now that we understand the concept, we can use some quick procedures to change from decimals to percent, and vice versa.

Changing a Decimal to Percent
1. Move the decimal point two places to the right.
2. Add the % symbol.

EXAMPLE 1 Change to a percent (a) 0.23 (b) 0.461 (c) 0.4

We move the decimal point two places to the right and add the % symbol.

(a) 0.23 = 23% (b) 0.461 = 46.1% (c) 0.4 = 0.40 = 40%

Be sure to follow the same procedure for percents that are less than 1%. Remember, 0.01 is 1%. Thus we would expect 0.001 to be less than 1%. 0.001 = 0.1% or one-tenth (0.1) of a percent.

EXAMPLE 2 Change to a percent (a) 0.0364 (b) 0.0026 (c) 0.0008

We move the decimal point two places to the right and add the % symbol.

(a) 0.0364 = 3.64% (b) 0.0026 = 0.26% (c) 0.0008 = 0.08%

Be sure to follow the same procedure for percents that are greater than 100%. Remember 1 is 100%. Thus we would expect 1.5 to be greater than 100%. 1.5 = 150%

EXAMPLE 3 Change to a percent (a) 1.48 (b) 2.938 (c) 4.5

We move the decimal point two places to the right and add the percent symbol.

(a) 1.48 = 148% (b) 2.938 = 293.8% (c) 4.5 = 4.50 = 450%

Changing from a Percent to a Decimal

In this procedure we move the decimal point to the left and remove the % symbol.

Changing a Percent to a Decimal

1. Move the decimal point two places to the left.
2. Remove the % symbol.

EXAMPLE 4 Change to a decimal (a) 16% (b) 143%

First we move the decimal point two places to the left. Observe in each case that the decimal point is not written but is understood to be to the right of the last digit in the percent. Then we remove the % symbol.

(a) 16% = 16.% = 0.16

(b) 143% = 143.% = 1.43

EXAMPLE 5 Change to a decimal (a) 4% (b) 3.2% (c) 0.6%

First we move the decimal point two places to the left. Then we remove the % symbol.

(a) 4% = 4.% = 0.04

(b) 3.2% = 0.032 (c) 0.6% = 0.006

Finding the Percent of a Number

How do we find 60% of 20? Let us relate it to a problem we did in earlier Section.

Consider the problem

What is 3/5 of 20?

5
3 #4
= 3% 26 = 12 The answer is 12.

Since a percent is really a fraction, a percent problem is solved in a similar way to solving a fraction problem. Since 3/5=6/10=60%, we could write the problem as

What is 60% of 20?

= 60% x 20
= 0.60 x 20
= 12.0 The answer is 12.

Thus we have developed a rule.

Finding the Percent of a Number

To find the percent of a number, merely change the percent to a decimal and
multiply by the decimal.

EXAMPLE 6 Find.

(a) 10% of 36 (b) 2% of 350 (c) 182% of 12 (d) 0.3% of 42

(a) 10% of 36 = 0.10 × 36 = 3.6 (b) 2% of 350 = 0.02 x 350 = 7

There are many real-life applications for finding the percent of a number. When you £0 shopping in a store, you may find sale merchandise marked 35% off. This means that the sale price is 35% off the regular price. That is, 35% of the regular price is subtracted from the regular price to get the sale price.

EXAMPLE 7 A store is having a sale of 35% off the retail price of all sofas. Melissa wants to buy a particular sofa that normally sells for $595.

(a) How much will Melissa save if she buys the sofa on sale?

(b) What is the purchase price if Melissa buys the sofa on sale?

(a) To find 35% of $595 we $595
will need to multiply x 0.35
0.35 x 595. 2975
1785
$208.25

Thus Melissa would save $208.25 if she buys the sofa on sale.

(b) The purchase price is $595.00
the difference — 208.25
between the original price $386.75
and the amount saved.

Thus Melissa bought the sofa on sale for $386.75.

Finding the Missing Percent When Given Two Numbers

Recall that we can write 3/4 as 75/100; or 75%. If we were asked the question, ‘‘What percent is 3 of 4?’’ we would say 75%. This gives us a procedure for finding what percent one number is of a second number.

Finding the Missing Percent
1. Write a fraction with the two numbers. The number after the word ‘‘of’’ is
always the denominator, and the other number is the numerator.
2. Simplify the fraction (if possible).
3. Change the fraction to a decimal.
4. Express the decimal as a percent.

A very common problem is the following:

EXAMPLE 8 What percent of 24 is 15?

This can be quickly solved as follows:

Step 1 15/24 Write the relationship as a fraction. The number after ‘‘of’’ is 24, so the 24 is in the denominator.

Step 2 = 5/8 Simplify the fraction (when possible).

Step 3 = 0.625 Change the fraction to a decimal.

Step 4 = 62.5% Change the decimal to percent.

The question in Example 9 can also be written as ‘‘15 is what percent of 24?’’ To answer the question, we begin by writing the relationship as 15/24. Remember ‘‘of’’ 24 means 24 will be in the denominator.

EXAMPLE 9 (a) 82 is what percent of 200? (b) What percent of 16 is 3.8? (c) $150 is what percent of $120?

(a) 82 is what percent of 200?

82/200 Write the relationship as a fraction with 200 in the denominator.

82/200 = 41/100 = 0.41 = 41%

(b) What percent of 16 is 3.8?

3.8/16 Write the relationship as a fraction.

You can divide to change the fraction to a decimal and then change the decimal to a percent.

0.2375—>23.75%
16)3.8000

EXAMPLE 10 Marcia had 29 shots on the goal during the last high school field hockey season. She actually scored a goal 8 times. What percent of her total shots were goals? Round your answer to the nearest whole percent.

Marcia scored a goal 8 times out of 29 tries. We want to know what percent of 29 is 8.

Step 1 8/29 Express the relationship as a fraction. The number after the word “‘of’’ is 29, so the 29 appears in the denominator.

Step 2 Note that this fraction cannot be reduced.

Step 3 = 0.2758... The decimal equivalent of the fraction has many digits.

Step 4 = 27.58...% We change the decimal to a percent, which we round to the nearest whole percent.

≈ 28%

Therefore, Marcia scored a goal approximately 28% of the time she made a shot on the goal.

Estimation

After studying this section, you will be able to:

1. Use rounding to estimate.

Using Rounding to Estimate

Estimation is the process of finding an approximate answer. It is not designed to provide an exact answer. Estimation will give you a rough idea of what the answer might be. For any given problem, you may choose to estimate in many different ways.

Estimation by Rounding :
1. Round each number to an appropriate place.
2. Perform the calculation with the rounded numbers.

EXAMPLE 1 Find an estimate of the product 5368 × 2864.

Step 1 Round 5368 to 5000.

Round 2864 to 3000.

Step 2 Multiply.

5000 x 3000 = 15,000,000

An estimate of the product is 15,000,000.

EXAMPLE 2 The four walls of a college classroom are 221/4 feet long and 83/4 feet high. A painter needs to know the area of these four walls in square feet. Since paint is sold in gallons, an estimate will do. Estimate the area of the four walls.

Step 1 Round 221/4 feet to 20 feet.

Round 83/4 feet to 9 feet.

Step 2 Multiply 20 × 9 to obtain an estimate of the area of one wall.

Multiply 20 × 9 × 4 to obtain an estimate of the area of all four walls.

20 × 9 × 4 = 720 square feet

Our estimate for the painter is 720 square feet of wall space.

EXAMPLE 3 Won Lin has a small compact car. He drove 396.8 miles in his car and used 8.4 gallons of gas.

(a) Estimate the number of miles he gets per gallon.

(b) Estimate how much it will cost him for fuel to drive on a cross-country trip of 2764 miles if gasoline usually costs $1.399/10 per gallon.

(a) Round 396.8 miles to 400 miles. Round 8.4 gallons to 8 gallons. Now divide.

400/8 = 50

Won Lin’s car gets about 50 miles per gallon.

(b) We will need to use the information we found in part (a) to determine how many
gallons of gasoline Won Lin will use on his trip. Round 2764 miles to 3000 miles and divide 3000 miles by 50 gallons.

3000/50 60

Won Lin will use about 60 gallons of gas for his cross-country trip.

To estimate the cost, we need to ask ourselves, ‘‘What kind of an estimate are we looking for?’’ It may be sufficient to round $1.399/10 to $1.00 and multiply.

60 × $1.00 = $60.00

Keep in mind that this is a broad estimate. You may want an estimate that will be
closer to the exact answer. In that case round $1.399/10 to $1.40 and multiply.

60 × $1.40 = $84.00

Caution : An estimate is only a rough guess. If we are estimating the cost of some- thing, we may want to round the unit cost so that we get closer to the actual amount. It is a good idea to round above the unit cost to make sure that we will have enough money for the expenditure. The actual cost of the cross-country trip to the nearest penny is $81.86.